1. Математический анализ (850924), страница 3
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( * . 6, , " + % * &,1 , , " *, * & , , { .2 & ( & x ! x0 O_ (x0 )). ? %, & &, * , , , , % . 3.2 ( ). 8x 2 O_ (x0) f (x) g(x) #xlim!x0f (x) xlim!x g(x)xlim!x0f (x) xlim!x g(x):00 xlim!x f (x) = Axlim!x g(x) = B , A > B . $" " > 0 , " "- A B ( . 3.4).? + * " < A ;2 B .
( ).0;. 3.4? " "9 _1(x0) : 8x 2 _1(x0) f (x) 2 "(A)9 _2(x0) : 8x 2 _2(x0) f (x) 2 "(B ):370* = min(1 2). 2 _ (x0) " _1(x0) _2(x0), % (", ) ( . 3.5 = 1).;. 3.5 + 8x 2 _ (x0) f (x) 2 "(A), g(x) 2 "(B ) , ,f (x) > g(x), . = , 1, A > B A B . 2 . 3.3. = f (x) < g(x) * . 6, f (x) = ;jxj, g(x) = jxj, f (x) < g(x) 8x 6= 0.. " O_ (0), , ( . 3.6),;lim f (x) = xlim!0 g(x) = 0:x!0. 3.6) , 3.2 .
6 * , f (x) < g(x) ) xlim!x f (x) < xlim!x g(x):00 3.3 ( - ./). 8x 2 _0(x0 ) '(x) f (x) Q(x)38(3:3)'(x) = xlim!x Q(x) = A x0 ./ f (x) - A:xlim!x0 0xlim!x0 f (x) = A:.8" > 0 9 _1(x0 ) : 8x 2 _1(x0 ) '(x) 2 "(A)9 _2(x0) : 8x 2 _2(x0) Q(x) 2 "(A):* = min(0 1 2).2 8x 2 _(x0) '(x) 2 "(A) Q(x) 2 "(A) (3.3). = , f (x) 2 "(A) 8x 2 _(x0) ( . 3.7), * x 2 _(x0) f (x) * * '(x) Q(x) (f (x) { * &).;.
3.72 ",8" > 0 9_ (x0) : 8x 2 _(x0) f (x) 2 "(A) ) 9 xlim!x f (x) = A:sin x . 3.1. 4 xlim!0 x R O ( . 3.8)6 AOB = x BC ?OA AD?OA0, ,OC = R cos x BC = R sin x AD = R tg x:39;. 3.8 ,: 4COB < : : AOB < : 4AOD: ! , 8x 2 0 2 :1 R2 sin x cos x < x R2 < 1 R2 tg x222 21 R2 sin x,cos x < sinx x < cos1 x " ,1 > sin x > cos xcos xx , (3:4)cos x < sinx x < cos1 x :$ "! & (3.4) + 8x 2 ; 2 0 , , 8x 2 _0(0) 0 = 2 .2 cos 0 = 1, " x = 0 cos x " 1 8x, " 0 ( & cos x), lim cos x = 1:x!0(@ " .
* R4.) ), ,1 = 1 = 1:limx!0 cos x 140E (3.4), (3.3) sin x = 1:(3:5)limx!0 x( # # ., # {!x1lim 1 + x = e:(3:6)x!+1 " 1 x = n" R1. ? , &!x 1 + x1 e * x ! +1 x ! ;1, , x ! 1.D 1 & , (3.6) x x1 x ! 1 x ! 0.1=x = e:lim(1+x)x!0(3:7) 3.4. & f (x) O_ (x0). P f (x) * x ! x0, 8M > 0 9_ (x0) : 8x 2 _ (x0) jf (x)j > M:2 *, " " % , & f (x) " " % x ! x0, x ! x0 xlim!x f (x) , , " :0xlim!x0 f (x)= 1: 3.2.
P f (x) = x12 " " % x ! 0. $ ( . 3.9),418M > 0 9_ (0) : 8x 2 _ (0) f (x) > M ) jf (x)j > M:1 = 1:limx!0 x2;. 3.9 3.3. P f (x) = ; x12 * " " % x ! 0. $ ( . 3.10),8M > 0 9_ (0) : 8x 2 _ (0) f (x) < ;M ) jf (x)j > M: 1!;;lim ; x2 = 1:x!0. 3.10 3.4. 4 & f (x) = x1 ( . 3.11). 3.1142? M > 0 _ (0) . 3.11.$ + 8x 2 _ (0) f (x) > M x > 0 f (x) < ;M x < 0.1 = 1, .. &= , 8x 2 _(0) jf (x)j > M ) xlim!0 x1f (x) = x " " % x ! 0.= " " % & " " %* . 3.5. 8 8M > 0 9_ (x0 ) : 8x 2 _ (x0) f (x) > M ( f (x) < ;M ) & f (x) " " % x ! x0 - ( /) %( xlim!x f (x) = ;1):) , 3.2 3.3, , 1!1lim = +1lim ; x2 = ;1:x!0 x2x! 04 " " % & 1 " x!limf (x) = 1 f (x0 + 0) = 1,x +0lim f (x) = 1 f (x0 ; 0) = 1.
= * " x!x ;0 " % * . 3.5. 4 & f (x) = p1x " x = 0. 2 % x > 0, " x = 0 * x ! 0 + 0. ( . 3.12), 1 = +1p8M > 0 9N : 8x 2 (0 N ) f (x) > M ) x!lim0+0 x.. & p1x { * " " % x ! 0+0.xlim!x0f (x) = +100043;;;. 3.12 3.6. x = x0 & & y = f (x), " f (x0 + 0) f (x0 ; 0) +1 ;1.S "* & y = f (x) x = x0 * ( . 3.13).. 3.13$ , f (x0 + 0) = +1, x ! x0 + 0 & y = f (x) "* x = x0 , " %, " { " x0. " * " * .<, % 3.2{3.5 x = 0 .$-+ ,&+1.
*: f (x) * x0 ( ) ! (. 3.1{3.3).2. 1* # ? (. 3.1).1?3. 6 xlimsin!0 x444. * % *: (. 3.2, 3.3).5. 1* #0 #? (. (3.5), (3.6), (3.7)).6. "* "5 *:, "* "5%, "* "5 : (. 3.4, 3.5).7. 4 * "* *:?8. 1* # 6 xlimf (x) x!;1lim f (x)?!1 f (x) 6 x!lim+19. * *: (. 3.6). =*# !* *: "# * (. . 3.13).1 6= 0, * ** " = 1 j sin 1 ; 0j < "3. ; 6. ;, xlimsin!0 xx1(k = 0 1 2 : : :) , , % * xk = +k2 x 2 _(0), ** " " * _(0) (xk 2 _(0) 1 = 1).
"5 jk j, sinxk+ ! ** ! *:sin x1 * x = 0.7. 1 2 . = "* *: * x0 !,*! *! 2 * "* "5 x ! x0.8. xlim!x0 f (x) = +1 ) xlim!x0 f (x) = 1 xlim!x0 f (x) = ;1 ) xlim!x0 f (x) = 1(. 3.2, 3.3), lim! f (x) = 1 6) xlim!x0 f (x) = +1 6) xlim!x0 f (x) = ;1x x0(. 3.4).45 4 89 456! " ;4 & & (y = f (x) y = g(x)), . 4.1, 4.2.. 4.1. 4.2= + & " x0. ,xlim!x f (x) = f (x0 ):06 & y = g(x) c " x0 . ) xlim!x g(x) 1 , g(x0), :0xlim!x0 g(x) = a 6= g(x0): 4.1. P f (x), - O(x0), x0, 9 xlim!x f (x)0 & f (x) + :xlim!x0f (x) = f (x0 ):(4:1)P f (x), %, x0, & g(x) () x0.? , + &ax loga x x sin x cos x tg x ctg x arcsin x arccos xarctg x arcctg x sh x ch x th x cth x46 * ?<. , , , lim cos x = cos 0 = 1:x!0() , . (3.5).)E (4.1) & f (x) x0 ( .
1.3) f (x0 ; 0) = f (x0 + 0) = f (x0 ):(4:2) % + x0 #& f (x). $ % (4.2) & 1".1. 8 f (x0 ; 0) = f (x0 + 0) 6= f (x0 ) 0 # & f (x). (< f (x0 ) * " .) 4.1. (x 6= 0Jf (x) = j1xj x = 0.6 . 4.3 & & f (x).;. 4.3 , f (0 ; 0) = f (0 + 0) = 0, f (0) = 1. = ,x = 0 { & f (x): <, * f (0) = 0, .472. 8 f (x0 ; 0) 6= f (x0 + 0) x0 # # & f (x).(< f (x0 ) * " ", * " .) 4.2. (x 0Jf (x) = xx+ 1 x < 0.;. 4.4< ( .
4.4) f (0 ; 0) = 1 f (0 + 0) = 0. = , x = 0 { & f (x). x = 0 & f (x) , " 1 x < 0 , 0 x = 0 , " 0 x > 0.3. , & f (x) # I . ) 1 f (x0 ; 0) f (x0 +0).$ # II . $ II " " 1 . 4.3. f (x) = 51=x (x 6= 0). <, 1 ;! ( ;1 x ! 0 ; 0,+1 x ! 0 + 0,x :f (0 ; 0) = 0 f (0 + 0) = +1:2 f (0 + 0) = +1, x = 0 { & f (x)II .48 & f (x) " x = 0 ( . & . 4.5). D" " & &, , 1) f (0 + 0) = +1 ) x = 0 { &J2) x ! 1 ) x1 ! 0 ) 51=x ! 1, 51=x > 1 x > 0 51=x < 1 x < 0:6 " , = 1 & &. $ xlim!151=x = 1 )( : (2:4))f (x) = lim 1 51=x = 0 1 = 0k = xlim!1 x x!1 x;1=xb = xlim!1 (f (x) ; kx) = xlim!1 5 = 1:y = kx + b ) y = 1 { ..
4.5# E (4.1), 1 & f (x) 0, xlim!x 0f (x) = f (x0 )8" > 0 9_ (x0) : 8x 2 _(x0 ) f (x) 2 "(f (x0 ))49(4:3)( . 3.3). 6 f (x) 2 "(f (x0 )) = 0 (f (x0 ) "- f (x0), (4.3) _ (x0) * "" (x0). $ 1 (4.1) "" ; ":P f (x) 0 , 8" > 0 9(x0) : 8x 2 (x0 ) f (x) 2 "(f (x0 ))(4:4) 4.1 ( # .-/). 1 - ./ y = f (u(x)). # ./ u(x) 0 # :xlim!x0 u(x) = * ./ y = f (u) # u = , -./ y = f (u(x)) 0 # f () ..xlim!x0 f (u(x)) =f (xlim(4:5)!x u(x)):E* * & ( &)..
6 & f (u) u = , 8" > 0 91() : 8u 2 1() f (u) 2 "(f ())(4:6) xlim!x u(x) = ) 81 > 0 9 _ (x0) : 8x 2 _(x0 ) u(x) 2 1()00 (4.6) u = u(x) (4:7)8x 2 _(x0) f (u(x)) 2 "(f ()):2 ", " > 0 1(), _(x0) (4.7), ..8" > 0 9_ (x0 ) : 8x 2 _ (x0) f (u(x)) 2 "(f (A)):50= ,xlim!x0f (u(x)) = f (A) .2 4.1 , , R3 ( . (3.5){(3.7)), * ,ln(1 + x) = lim ln(1 + x) x = ln lim (1 + x) x = ln e = 1:limx!0x!0x!0x< & ln(1 + x)1=x * &:u = (1 + x)1=x { &, y = ln u { % &.1=x = e ( . (3.7)), & ln u u = elim(1+x)x!0 + &.
E 4.1, * &.2 ",ln(1 + x) = 1:(4:8)limx!0xx;1e4 xlim!0 x :8 y = ex ; 1, x = ln(1 + y) , , y ! 0 x ! 0, x;1y = lim1 = 1 = 1:e=limlimy!0 ln(1 + y ) y!0 ln(1 + y )x!0 x1y),x;1elim= 1:(4:9)x!0 x11 4.2 ( # - ./). ./u(x) # 0, ./ f (u) # u0 = f (x0 ), - ./ f (u(x)) # 0 .51.(4.1), xlim!x0 4.1 f (u(x)) = f (xlim!x u(x)) = f (u(x0 ))0 , (4.1), & f (u(x)) 0. 4.3( . / ##./). ./ f (x) g (x) ## 0 , { , 0 ## ./ f (x) + g (x)J Cf (x)JCg(x)J f (x)g(x) , g(x0) 6= 0, ./ fg((xx)) ..
, xlim!x0 (f (x) + g(x)) = xlim!x0f (x) + xlim!x g(x) = f (x0 ) + g(x0)0 , f (x) + g(x) 0. =% &.$ % + &. + & * % , * " .$ "1 , )# ./ &,1 + & 1 & " *&. ) 4.3 4.2, , , + & ( ) * * " . 4.4 ( # ./). ./ f (x) # x0 f (x0 ) 6= 0, f (x) 6= 0 x0 , f (x) ) f (x0 ).$" " > 0 , " "(f (x0 )) * ( . 4.6)..52;.
4.62, (4.4), 9(x0) : 8 x 2 (x0)f (x) 2 "(f (x0 )) * .6 (4.1) (4.4) % 1 &. & y = f (x) (x0), Tx = x ; x0 { 1 , Ty = f (x) ; f (x0 ) { 1 1&.2, , xlim!x0 f (x)= f (x0 ) () xlim!x Ty = 00x ! x0 () Tx ! 0 " &y = f (x) 0 lim Ty = 0:x!0(4:10)$-+ ,&+1. *: * (. 4.1 (4.1), (4.2), (4.10)).2. % *:, *, 2 *?3. % *:, * *, " 2 *?4. + * # 06 (. . 1, 2,3).5.
* #* *:(. 4.1).6. 1* *: #0 2? 60 ** ? (. 4.2, 4.3).7. % *: f (x) " % * x0 : * " .532. ; % (. 4.1).3. ; % (. 4.1).7. %, *: f (x) # * x0. ; %, * x0 (. 4.4 " #* *:).54 5 456, 88; "$ & f (x) x0 ( . 4.1):xlim!x f (x) = f (x0 ) . 8 , #. 5.1. & f (x) x0. 8 09 x!limf (x) = f (x0 )x +00 & f (x) x0 9 x!limf (x) = f (x0 )x ;00 & x0.6 & *. 6, & # , * .@ * & . 5.2.
P f (x) La b], :1) (a b)J2) aJ3) b. ": f (x) 2 C La b]: 5.1. P, La b] * " a b:55;. 5.16, . 5.1 & & y = f (x), La b], & a, b ( + % "" ).$ , "P, , " , & . 5.1( ./). ./f (x) 2 C La b] / La b] , -# , , , 2 (a b) ;f ( ) = 0:. 5.2? .8 , & &,1 , " Ox( . 5.3).56;. 5.3<, & f (x) La b], * * f (a) f (b)& f (x) * "1 (a b) & * x ( . 5.3).2 5.1 % "* % + , *, *1 . 5.1. x3 + x2 ; 1 = 0:P f (x) = x3 + x2 ; 1 f (0) = ;1 < 0 f (1) = 1 > 0:< L0 1] & f (x) ,* .
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